Understanding Total Derivative: Step-By-Step Guide

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SUMMARY

The discussion focuses on the total derivative in the context of the chain rule, specifically illustrated through the equation D(p(a)) = S(p(a), a). The total derivative with respect to 'a' is expressed as dD(p(a))/dp * dp/da, which equals the sum of the partial derivatives of S with respect to p and a, multiplied by dp/da. This highlights the application of the chain rule in differentiating composite functions.

PREREQUISITES
  • Understanding of calculus, specifically differentiation and the chain rule.
  • Familiarity with partial derivatives and their notation.
  • Basic knowledge of functions and their compositions.
  • Experience with mathematical notation and manipulation.
NEXT STEPS
  • Study the application of the chain rule in multivariable calculus.
  • Learn about partial derivatives and their significance in optimization problems.
  • Explore examples of total derivatives in real-world applications.
  • Investigate the relationship between total derivatives and differential equations.
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Students and professionals in mathematics, physics, and engineering who are looking to deepen their understanding of calculus, particularly in the context of derivatives and their applications.

Tom McCurdy
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Hi I was hoping someone could explain how this works to me

I ran across this example in a book

D(p(a)) = S(p(a),a)
Total derivative with respect a

[tex]\frac{dD(p(a))}{dp} \frac{dp}{da}= \frac{\partial S(p(a),a)}{\partial p} \frac{dp}{da} + \frac{\partial S(p(a),a)}{\partial a}[/tex]
 
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It's just the chain rule.
 

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